Many possible choices of \(\tilde{g}\)  exist. In [1]}, [2]}, \(\tilde{g}\)  is an \((M-1)\) -degree polynomial expressed in Lagrange basis, defined on collocation points (CPs) \(\xi :=\lbrace \xi _k\rbrace _{k=1}^M\)  computed as Gauss-Hermite quadrature nodes, i.e.:
\(\tilde{g}(x):=\sum _{k=1}^M a_k \ell _k(x), \qquad \ell _k(x):=\prod _{\begin{array}{c}1\le j\le M\\j \ne k\end{array}} \frac{x-\xi _j}{\xi _k-\xi _j}, \quad k=1,\dots , M,\) 
